Diapositiva 1

1
Trento PhD course on Ultracold atomic Fermi
gases (February-March 2008)
Lecture 4
Ideal Fermi gas and the role of the interaction
Sandro Stringari
University of Trento
CNR-INFM
2
Piano del corso
Lecture 2. BEC gases and the role of the
interaction The Gross-Pitaevskii equation
Lecture 3. Bose-Einstein condensates in harmonic
traps
Lecture 4. Ideal Fermi gas and the role of the
interaction
Lecture 5. BEC-BCS crossover and the Bogoliubov
de Gennes equations
Lecture 6. Interacting Fermi gases in harmonic
traps
Lecture 7. Superfluidity and hydrodynamic
behavior
Lecture 8. Rotating Fermi superfluids
Lecture 9. Spin polarized Fermi gases
Lecture 10. Fermi gases in periodic potentials
3
Quantum statistics and temperature scales
4
Fermi energy in harmonic trap
If one can use
semiclassical approximation
distribution function
- Accounts for Pauli exclusion principle -
Violates Heisenberg uncertainty relation
(semiclassical)
5
3D density of sp states (dimensionality fixes
dependence)
normalization can be written as
At T0
and integration yields

Same dependence on N and on trapping frequency
as for BEC critical temperature !
6
Alternative calculation of Fermi energy
Fermi energy is defined by highest
energy level occupied at T0. In 3D isotropic
potential ( ) each
level has degeneracy (n1)(n2)/2 where n is
principal number ( )
Total number of atom is given by
(for large N)
semiclassical approx.
Since for large one can write
one recovers result for Fermi energy
Result scales on N as critical temperature for
BEC
(general condition of quantum degeneracy)
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Comparison with interacting Bose-Einstein
condensates (T0)
Density profiles look similar in both cases
size is larger than oscillator length and
increases with N (quantum pressure replaces
role of interactions
ideal FERMI
interacting BEC
p-integration of distribution function yields
Thomas-Fermi density
LDA approximation (Thomas-Fermi) yields (

)
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• Atomic momentum distributions differ in deep way
• anistropy in BEC vs isotropy in Fermi gases
• In BEC momentum width decreases with N
• in Fermi gases momentum width increases with N

ideal FERMI
interacting BEC
r-integration of distribution function yields
momentum distribution
Fourier transform of condensate wave function
yields (
)
anisotropy
isotropy
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Release energy
energy of the gas measured after releasing the
trap
For non interacting Fermi gas release energy
coincides with kinetic energy. For harmonic
trapping one has . total
energy as a function of T Approaches finite
value as Approaches for large T
Behaviour of
release energy differs significantly from
classical gas as well as from Bose gas

ideal BEC (interacting BEC has finite
small value at T0)
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Experimental evidence of Fermi degeneracy
in trapped atomic gases
Measurement of release energy reveals increase
of respect to unity. Direct
signature of quantum pressure effects (De Marco
and Jin 1999)
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- Ideal gas is accurate model for spin polarized
Fermi gas (consequence of Pauli principle).
Unique exp. possibility of realizing text-book
conditions (Bloch oscillations, insulating vs
conducting behaviour ) with challenging
applicative perspectives (sensors)
Example Spin polarized Fermi gas performing
Bloch oscillation in the presence of periodic
potential (G. Roati et al. PRL 92, 230402 (2004)

• d is periodicity of lattice
• Bloch frequency measured with10(-4)
• precision in this experiment

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In two spin species Fermi gas s-wave
interaction can be strongly active
INTERACTIONS
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Interactions can be tuned thanks to availability
of Feshbach resonance
we assume braod resonance

• atoms collide in open channel at
• small energy
• same atoms in different hyperfine states
• form a bound state in closed channel
• coupling through hyperfine interactions
• between open and closed channel
• if open and closed channel have different
• magnetic moment ? magnetically tunable
• ?E ? ? ? B

no bound state
Resonance when bound-state and continuum become
degenerate
bound state
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Feshbach resonance and scattering amplitude
When scattering length is positive the
interaction can give rise to weakly bound
molecules of size a. If size
of molecules is much smaller than average
distance between molecules and formation of
molecules reduces to 2-body problem (difference
with respect to Cooper pairs in BCS theory).
Simplest description of 2-body problem based on
scattering amplitude f (wave function of
relative motion of two scattering particles).
At small k s-wave scattering is leading effect
and one can write
scattering amplitude
is effective range of the force
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Some properties of the scattering amplitude
( s-wave
scattering length)
(unitary limit, result holds if
broad resonance
If and scattering
amplitude has pole on imaginary axis at
.
Pole defines molecular bound state with
energy Wave function takes form (molecular
size fixed by scattering length)
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Feshbach resonance
Binding energy of potassium molecules measured at
Jila (Regal et al. 2003)
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Interactions and many-body effects
For dilute and cold gases one can use effective
interaction
based on s-wave scattering coupling constant
Scatttering length only interaction length of
the problem (range of the force assumed to be
much smaller than interparticle distance)
Perturbative estimate of interactions in
harmonically trapped gas. Assume same trapping
for both spin species and ( is
total density of the gas)
unpeturbed TF profile
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Use of unperturbed density profile yields result

geometrical average of trapping frequencies
dimensionless interaction parameter

gas parameter
Example

• Questions
• Comparison with perturbation theory applied to
  BEC
• Validity of perturbation theory

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Interaction effects behave quite differently in
Fermi and BEC gases. In a BEC gas use of
perturbation theory yields
BEC gas (perturbation theory)

• This ratio is gtgt1 in most physical systems
  because of the large value of
• Thomas-Fermi parameter
• - In Fermi gas the ratio depends on the
  combination
• Perturbation theory fails in BEC gases
• This explains why Gross-Pitaevskii theory is
• non trivial even if it is basically mean field

example
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Validity of perturbation theory in ultracold
Fermi gases

• Perturbation theory predicts that interaction
  effects
• are small if is small
• Always true?
• What happens if ?

Contrary to predictions of perturbation theory,
interactions can deeply modify the nature of a
dilute Fermi gas even if

• alt0 system is superfluid at low temperature
  (BCS superfluidity)
• agt0 atoms can form bound molecules (giving
  rise to BEC at low T)

In both cases deep modification of many-body wave
function. Ideal Fermi gas is no longer proper
starting point
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Many-body aspects (BEC BCS crossover)
BEC regime
BCS regime
unitary limit
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BCS theory (alt0)

• Theory of superfluidity for and
  is well established
• (BCS theory). Key results for uniform gases
• Pairing between spin-up and spin-down atoms in
  momentum space
• (Cooper pairing)
• Order parameter characterizes long
• range order of 2-body density matrix.
• Quasi-particle excitation spectrum
• is gapped. Gap is related to order
• parameter by relationship
• BCS critical temperature
• BCS theory is valid only if
• Example
  . BCS critical temperature is too low
• even smaller than oscillator temperature (
  ) with ).
  
  

(Gorkov, Melik-Barkhudarov, 1961)



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BEC gas of molecules (agt0) calculation of
Condition for having dilute molecular gas
requires that distance between molecules be
much larger than molecular size Such a
molecular gas behaves like a dilute gas of
bosons and exhibits BEC
Value of critical temperature for a gas of
bosons of mass 2m (molecules) at density
is directly related to value of Fermi energy
of fermions of mass m at the same density n.
Relationship is even more favourable for harmonic
trapping where and
yielding
Critical temperature for superfluidity is much
higher in BEC rather than in BCS side where it is
exponentially small.
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Some questions concerning the BEC-BCS crossover

• search for many-body theory describing the
  transition
• interaction between molecules on BEC side
• lifetime of molecules
• what happens at unitarity ?
  (neither molecules, nor Cooper pairs)
• can we describe dynamics ?
• can we probe superfluidity ?
• can we achieve the low T values required by BCS
  regime ?

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• Need for a many-body theory describing the
  transition between
• various regimes of interacting Fermi gases,
• including BEC, BCS and unitary regime
• Bogoliubov de Gennes mean field theory
• (approximate, but qualitatively correct)
• Ab initio Monte Carlo simulations (in principle
  exact)